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Subjects

Abstract

Maxwell’s demon is an imaginary entity that reduces the entropy of a system and generates free energy in the system. About 150 years after its proposal, theoretical studies explained the physical validity of Maxwell’s demon in the context of information thermodynamics, and there have been successful experimental demonstrations of energy generation by the demon. The demon’s next task is to convert the generated free energy to work that acts on the surroundings. Here, we demonstrate that Maxwell’s demon can generate and output electric current and power with individual randomly moving electrons in small transistors. Real-time monitoring of electron motion shows that two transistors functioning as gates that control an electron’s trajectory so that an electron moves directionally. A numerical calculation reveals that power generation is increased by miniaturizing the room in which the electrons are partitioned. These results suggest that evolving transistor-miniaturization technology can increase the demon’s power output.

Introduction

One famous example of energy generation by Maxwell’s demon is when it observes individual randomly moving gas particles in a box1,2,3. As feedback responding to the observation, the demon opens and closes a gate separating the box into two areas so that colder and hotter gas particles are sorted out. Thanks to such observation and the following feedback, the temperature in one area differs from that in the other, which means free energy is generated and entropy is reduced in the box. The noteworthy feature of this energy generation is that, without friction, Maxwell’s demon does not consume energy during the feedback. This gives rise to a question regarding a violation of the second law of thermodynamics because the demon looks like it is producing energy from a thermal bath without any dissipation.

This question has been answered by the concept of information thermodynamics4,5,6,7,8,9,10,11,12,13: The generated free energy originates from the demon’s memory. The demon uses energy during a cycle of the memory operation for storing the information about randomly moving objects at the measurement and erasing it after the feedback4,5. Consequently, the energy it generates perfectly follows the second law of thermodynamics: When Maxwell’s demon generates energy, a corresponding dissipation takes place in the demon’s memory.

In addition to this theoretical progress, technical advances allowing access to a small system in which a few small objects are thermally fluctuating has led to experimental demonstrations of Maxwell’s demon14,15,16,17,18,19,20,21,22,23, by using rotaxane molecules14 and beads a few-hundred nanometers in diameter15. Using electrons, the demon has also realized a Szilard's engine that generates free energy18. Further, an autonomous Maxwell’s demon that enables automatic observation of and feedback to electrons has been recently demonstrated using a chip integrated with superconductor devices, and the thermodynamics of the demon itself was revealed20. In addition, some previous experiments have succeeded in extracting work15,20 and power22 with the demon. Here we present a demonstration of one of the most natural possible Maxwell’s demon that outputs electric power by rectifying randomly moving electrons in transistors.

Results

Device structure and the demon’s feedback protocol

The power generation by the demon is carried out on a silicon transistor chip integrating a small box (hereafter referred to as a single-electron box (SEB)) and a charge sensor that monitors electron motion (Fig. 1a,b, also see Methods section and Supplementary Note 1). The SEB is separated from the source and drain by two transistors, G1 and G2, which function as gates for electrons. The number of electrons n in the SEB is around 100 on average. It fluctuates due to thermal agitation (Fig. 1c), which is monitored in real time with the sensor used for the feedback, which is explained next.

(a) False-color scanning electron microscope image and (b) schematics of the fabricated device. Above a silicon wire channel covered with a silicon dioxide, two gate terminals are formed to make field-effect transistors Gate(G) G1 and G2. The single-electron box (SEB) is electrically formed between the source (S) and drain (D) because G1 and G2 form energy barriers between the S and D as shown in d. Another channel functioning as a charge sensor is located close to the SEB. Above the whole area, there is an upper gate (UG), to which positive voltage VUG is applied. The details are explained in Methods section. (c) Single-electron detection at room temperature. Current flowing through the sensor is monitored and the abrupt reduction (increase) of current with the same step height means that a single-electron enters (leaves) the SEB. k is the deviation from the average of the number of electrons n in the SEB. We applied VUG=3.05 V, VED=1 V, VG1=−2.4 V, VG2=−1.95 V and VS=VD=−0.6 V. (d) Schematics of current generation by rectifying randomly moving electrons. The four upper and lower illustrations are schematics and energy-band diagrams along the S, SEB and D, respectively. For simplicity, they show how one electron is transferred, meaning Δn explained in the main text is one. We define state A(B) as the case that G1 opens (closes) and G2 closes (opens). When G1 opens (closes), its energy barrier lowers (rises) and an electron shuttles between the S and SEB faster (slower).

When G1 opens and G2 closes, electrons shuttle randomly between the source and SEB due to thermal fluctuation (hereafter referred to as state A). When n increases by Δnthresh larger than zero from n observed initially at state A, G1 and G2 are closed and open, respectively, and then electrons shuttle randomly between the drain and SEB (state B). When n decreases by Δnthresh from n observed initially at state B, G1 and G2 are again open and closed, respectively (state A). This feedback cycle comprising state A and B transfers electrons from the source to drain (Fig. 1d). In other words, the feedback cycle rectifies the random motion of electrons and converts it into electric current IMD (= × e/Δtcyc, where 〈Δnt〉 is the ensemble average of the number of transferred electrons, e is the elementary charge and Δtcyc is the time taken for one cycle in which the state changes from A to B and back to A). Here, IMD is defined as a current flowing from the drain to source (forward current). As can be easily expected, when Δnthresh is negative, current flows in the opposite direction (backward current). When source–drain bias voltage VSD (= VS−VD) is applied across the source and drain, the electric power the demon generates can be evaluated to be −IMD × VSD, where the minus sign represents the current direction against VSD, or electrons climbing up the chemical potential.

Our silicon chip has two unique features that affect the demon’s performance. The first feature is that the transport of electrons between the SEB and source or drain shows directionally asymmetric characteristics24 because the transport is dominated by thermal hopping over the energy barrier induced by the transistor’s gate (Fig. 2a). The transition rates for this thermal hopping is tunable by the gate voltage and proportional to exp(−ΔE/kBT), where kB is Boltzmann’s constant, T is temperature and ΔE is the height of the energy barrier the electrons have to surmount. For electrons in the source and drain, ΔE is the difference between the top of the energy barrier and chemical potential μS and μD of the source and drain, respectively. Therefore, the transition rate for the electrons to enter the SEB is constant because voltages applied to the transistors’ gates, source and drain are constant during the measurement of electron motion. On the other hand, for electrons in the SEB, ΔE is the difference between the top of the energy barrier and the energy level of an electron in the SEB. This energy level is given by μSEB+(k−0.5)e/C, where μSEB and C are the chemical potential and capacitance of the SEB, respectively, and k is the deviation of n from the integer part of the average of thermally fluctuating n. Since the energy level varies according to k, the rate at which an electron leaves the SEB varies with k (Fig. 2b, the details are explained in Supplementary Note 3)24. Therefore, electrons move in a directionally asymmetric way, which is beneficial for power generation by the demon as explained later.

Figure 2: Directionally asymmetric electron transport in transistors.

(a) Energy-band diagram along the S and D, where μS, μD and μSEB are chemical potentials at the source (S), drain (D) and single-electron box (SEB). For simplicity, we consider the electron shuttling between the S and SEB. Γ+ and Γ− are transition rates from the S and SEB, respectively. ΔE is energy barrier height when an electron in the S surmounts the energy barrier. (b) Transition rates Γk+ and Γk− as a function of k, which is the deviation in n from the integer part of the average. They are the transition rates for electrons to enter (+) and leave (−) the SEB at k. The dotted lines show expected values. The details are explained in Supplementary Note 3. Since the number of data points at |k|>4 is relatively small, there are large errors at |k|≥4. (c) Current across the SEB ISEB generated by periodic switching without feedback, estimated with single-electron counting statistics, as a function of the shift in μSEB (ΔμSEB) when the state is switched from A to B. The upper axis depicts ΔVUG, which is the difference in voltage applied to the upper gate (UG) when the state is switched from A to B. State A and B are alternately switched every 4 s. Current is given by Δne/8, where Δn is the number of electrons transferred from S to D when the state is switched from A to B. The factor of 8 originates from the time it takes for the state changes from A to B and back to A, that is, 2 × 4 s. The details, including the evaluation of ΔμSEB, are explained in Supplementary Note 4.

The second unique feature of our chip is that undesirable work applied to the system during the feedback process can be eliminated. One of the most important conditions for the feedback from the demon is that no work be applied to observed objects, but it is difficult to meet this condition in experiments because of the difficulty in partitioning the objects by a gate. Indeed, in previous reports15,18,19,20, work was applied to the objects and then carefully considered as a net value to discuss information-originated free energy. In our chip, since transistors partition electrons using gates as mentioned above, the demon can, in principle, perform the feedback process without work exertion. However, in reality, since the transistors tuned by applied voltage VG1 and VG2 are capacitively coupled to the SEB, the change in VG1 and VG2 at the switch between state A and B shifts the SEB’s chemical potential μSEB, and thus an electron in the SEB can gain or lose energy. This energy shift, ΔμSEB, generates current even without the feedback by the mechanism of a single-electron ratchet25 (the details are explained in Supplementary Note 4). To eliminate ΔμSEB, in our operation scheme, we cover the whole area with another gate (hereafter referred to as an upper gate (UG)). The UG is capacitively coupled to the SEB, which enables us to control μSEB at state A and B separately. When state A and B are alternately switched periodically without the feedback26, even when voltage between the source and drain is zero, current flows due to the ratchet mechanism. This current can be tuned by tuning the UG-voltage difference ΔVUG between state A and B, that is, ΔVUG=VUG,A−VUG,B, where VUG,A(B) is UG voltage at the state A(B) (Fig. 2c). With the difference ΔVUG=0.4 V, no current is generated with the change between state A and B without feedback, which means that the UG enables us to eliminate undesirable work in the demon’s feedback and monitor pure current generated by the demon.

Power generation by Maxwell’s demon

All measurements were carried out at room temperature. The time interval Δtm of each measurement was set to 60 ms, which is short enough to monitor electron motion or k. More importantly, from the viewpoint of the demon’s performance, Δtm was a bit shorter than (or comparable to) the average time intervals for n to increase by Δnthresh at state A and decrease by Δnthresh at state B, which leads to high power generation and relatively high efficiency for information-to-energy conversion as discussed later. Note that k and Δnthresh are different: k is the deviation in n from the integer part of the average of n (k=n−nave, where nave is the integer part of the average of n) and Δnthresh is the deviation in n from the initially observed n in the state and the threshold value to perform feedback. In our experiment, Δnthresh is determined by threshold value of Idet. We set the threshold as 1.5δIdet, where δIdet is the change in Idet induced by the motion of an electron. In this work, since Idet showed discrete steps (Fig. 1c), the change in Idet became larger than 1.5δIdet when Δnthresh=2.

With the feedback, even when VSD=0 V, IMD of about 20 zA flows from the drain to source (forward current) (Fig. 3). On the other hand, no current is generated without the feedback (Fig. 2c). Current in the opposite direction (backward current) can also be generated when Δnthresh=−2. In addition, even when negative (positive) VSD is applied to the drain, forward (backward) current counter flows from the drain (source) to source (drain). These results indicate that the demon generates current by rectifying randomly moving electrons. Also noteworthy is that Monte-Carlo simulation with consideration of the directionally asymmetric electron shuttling mentioned above (Supplementary Note 2) can reproduce the experimental results well, which means that the directionally asymmetric electron shuttling actually takes place in our silicon chip.

Figure 3: Current generation by Maxwell’s demon as a function of the source–drain bias voltage VSD (=VS−VD).

Open dots are experimental results. The solid lines are data obtained from Monte-Carlo simulation, whose details are explained in Supplementary Note 2. The two insets show schematics for forward current (drain-to-source current obtained with Δn=2) and backward current (source-to-drain current obtained with Δn=−2) when negative VSD is applied. The reason the characteristics between forward and backward currents are asymmetric originates from the difference in the terminal from which electrons are supplied: In the forward direction, electrons are supplied from the S, whose chemical potential is fixed; in the backward direction, electrons are supplied from the D, whose chemical potential is varied by VSD. In the experiments, we applied VUG=3.1 V, VG1=−2.4 V and VG2=−1.95 V for state A; VUG=2.7 V, VG1=−3 V and VG2=−1.6 V for state B; VED=1 V, VS=−0.6 V for the both states.

Generated power is given by −IMD × VSD. When VSD is changed from 0 V in a negative direction, the generated power increases because of the increase in VSD and then decreases because of the reduction in IMD (Fig. 4a). This behaviour can also be reproduced by Monte-Carlo simulations. In our experiments, maximum power of about 0.5 zW was generated. Such small power and current can be evaluated thanks to real-time electron counting by the high-sensitivity charge sensor, which leads to this proof-of-concept demonstration of Maxwell’s demon. On the other hand, the generated power can be increased, in principle, by reducing Δtm and the gate-tunable transition time for electrons to hop over the energy barrier. However, from the technical perspective, Δtm is limited by the time resolution of the charge sensor. High-speed charge sensing, for example, using a reflectometry method based on radio-frequency signals27 or a faster feedback process using an on-chip circuit, can reduce Δtm and thus increase IMD and the power.

Figure 4: Power generation by the demon.

All voltages are the same as in the case in Fig. 3. (a) Generated power as a function of VD. The solid lines are data obtained from Monte-Carlo simulation. (b) Efficiency of information-to-energy conversion as a function of VSD. (c) Δtm dependence of the power and efficiency in Monte-Carlo simulation. (d,e) Numerical calculations of power generated by Maxwell’s demon. Generated power as a function of (d) VSD at various EC and (e) EC at various VSD. For all calculations, kBT=26 meV, Γ1=1 s−1, Γ2=10−10 s−1 and Δn=1 (Supplementary Note 5).

Efficiency of the demon’s power generation

The efficiency is defined as the ratio of the generated energy to the mutual information, which represents conversion efficiency from information to energy. The efficiency of information-to-energy conversion is given by , where ΔF is the change in the free energy and I is the mutual information, and depicts the mean per one measurement. When Δnt electrons are transferred from the source to drain, ΔF becomes ΔnteVD. When the number of measurement points for this electron transfer is Nt, is given by . When there is no error in the measurements of n, I equals Shannon entropy H given by , where Pi is the probability for the ith event. In our experiments, we use information about n: n is either increased by Δnthresh (decreased by Δnthresh) or not at state A (state B). From the probabilities experimentally obtained, Shannon entropies HA and HB of the measurement outcome at state A and B, respectively, are estimated. When the number of the measurement points at state A and B are NA and NB, respectively, H is given by (NAHA+NBHB)/(NA+NB). We assume I=H, or neglect errors in measurement of n. Consequently, since , we can estimate the lower bound of the efficiency based on electron counting statistics in the experiment (Supplementary Note 6).

The efficiency increases with |VSD| and reaches about 18% at the VSD that provides the maximum power (Fig. 4b). This is because the increase in |VSD| increases the energy gain of an electron transferred from the source to drain. The efficiency also depends on Δtm: When Δtm becomes longer, the efficiency increases and then saturates (Fig. 4c). The point of this behaviour is that it reflects how much of the information obtained by each measurement is used for the feedback: Measurements in the absence of the feedback waste the obtained information and reduce the efficiency. When Δtm becomes shorter (longer) than the time taken for n to increase or decrease by Δn at state A or B, respectively, the number of measurements without the feedback increases (saturates), which leads to a decrease (saturation) in efficiency (Fig. 4c; the details are explained in Supplementary Note 5). Compared to the simulated efficiency (24%) at Δtm of 60 ms (Fig. 4c), the efficiency of 18% obtained in our experiment at VSD=−34 mV is reasonably large. The further increase of |VSD| also increases the number of measurements in the absence of the feedback, which leads to reduced efficiency. Due to these features, the efficiency cannot reach 100% like it can in an ideal Szilard’s engine utilizing all of the information. On the other hand, the power decreases with increasing Δtm because the time interval for carrying out the feedback increases with Δtm, which reduces IMD. Therefore, there is a tradeoff between the efficiency and power against Δtm and, in this work, we adjusted it to 60 ms to ensure the reduction in the power is small and the efficiency is relatively large. Note that the origin of the power reduction is the missed detection of fast electron transitions within Δtm. Monte-Carlo simulation indicates that the power reduction caused by detection errors is <10% in this work.

Discussion

Numerical calculations indicate another way to increase the power generated by the demon (details are explained in Supplementary Note 2). For simplicity, we assume infinitely fast sensing and feedback and Δtm=0. In the case of Szilard’s engine, the generated energy is given by kBTln2, which is proportional to temperature. In our feedback with transistors, in which electrons are transported by thermal hopping, the generated power depends on another factor, which is the charging energy (EC=e2/2C) of the SEB. The power increases monotonically with EC (Fig. 4c,d). There are mainly two reasons for this behaviour of the generated power. The first is that the electron can gain energy multiples of 2EC when it enters the SEB from the source, which allows the electron to reach the drain highly biased and thus to gain larger energy. The second is related to how often this energy gain happens at larger EC. As explained above, the transition rate for electrons in the source to enter the SEB is independent of EC. This is in contrast to an SEB sandwiched by tunneling barriers: Larger EC reduces this transition rate due to the Coulomb-blockade effect, which leads to a reduction of power in spite of larger energy gain. In other words, Maxwell’s demon exploits transistors’ benefits as an increase in power by increasing EC, that is, decreasing C, which can be achieved by miniaturization technology for current transistors. We should note that large EC shortens the time that an electron stays in the SEB and tends to require faster sensing and feedback. At Δtm=60 ms, EC should be smaller than 35 meV to monitor the desired electron motion and perform the feedback before the electron’s departure from the SEB; otherwise, n changes before the feedback is completed.

The transistors provide a benefit to the generated power in this work and also have other merits. Since transistors can act as gates partitioning electrons, which have not been achieved with other systems, the procedure for the demon’s power output is relatively simpler than in previous work. In addition, with an electrical approach, the transistors can not only control the chemical potential and size of the box28,29, in which an electron is confined, but also form tunnel barriers30 and a couple of dots31,32. Operation in a wide range of temperatures, including room temperature as in this work, also deepens the analysis and applications of Maxwell’s demon. These features promise a experimental platform on which the demon plays active roles, such as a quantum Szilard engine33, noise squeezer21,34 and nano-scale heat engine35,36,37,38. Transistors also have another perspective supported by nanotechnology, which continues to make commercially available transistors smaller and smaller. Further shrinkage of transistors will further improve the controllability of electron motion and charge sensitivity and increase EC, all of which will lead to improvement in the demon’s performance. Therefore, we believe that Maxwell’s demon in transistors continue to improve in performance and find new applications.

Methods

Device fabrication and structure

The device is fabricated from a silicon-on-insulator (SOI) wafer. First, silicon nanowire channels for a SEB, source, drain and sensor are formed on an SOI layer with boron concentration of 1015 cm−3, followed by thermal oxidation. The width and thickness of SOI channels for the SEB and sensor are around 30 and 20 nm, respectively. Oxide thickness is 38 nm. Then, two gates, G1 and G2, composed of poly-crystalline silicon are formed on the SOI channel between the SEB and source/drain, followed by oxidation. G2 is designed to be larger than G1 to reduce capacitive coupling between the SEB and drain and suppress changes in the chemical potential at the SEB when voltage applied to the drain is changed. However, there is a risk that small electron traps are unintentionally formed under a large gate due to the structural fluctuation of the transistor channel. Therefore, we use a small G1 to mitigate the risk and a larger G2 to reduce the potential change. The width of G1 and G2 are 30 and 200 nm, respectively. Then the whole area is covered with a 50-nm-thick oxide interlayer formed by a chemical vapour deposition. Finally, another gate (UG) is formed on the whole area. The UG is used to control the chemical potential of the SEB, induce electrons in the source and drain, and to control current flowing through the channel of the sensor.

Data availability

The data that support the findings of this study are available from the corresponding author on request.

Contributions

K.C. and K.N. conceived the experiments, analysed the data, performed the simulations and prepared the manuscript. K.C. and S.D. performed the measurements. K.N. fabricated the devices and performed the numerical calculations. All authors discussed the results and commented on the manuscript. A.F. supervised the project.

Supplementary information

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